Radical layer length and syzygy-finite algebras

Let $\mathrm{\Lambda }$ be an artin algebra. We obtain that $\mathrm{\Lambda }$ is syzygy-finite when the radical layer length of $\mathrm{\Lambda }$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the category $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$ of finitely generated right $\mathrm{\Lambda }$-modules in terms of the projective of certain class of simple right $\mathrm{\Lambda }$-modules and also get the left big finitistic dimension conjecture holds.